L_{1}-estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup

Hendrik Vogt^{∗}

Abstract

We investigate selfadjoint positivity preserving C_{0}-semigroups that are dominated by the free heat semigroup on ℝ^{d}. Major examples are semigroups generated by Dirichlet Laplacians on open subsets or by Schrödinger operators with absorption potentials. We show explicit global Gaussian upper bounds for the kernel that correctly reflect the exponential decay of the semigroup. For eigenfunctions of the generator that correspond to eigenvalues below the essential spectrum we prove estimates of their L_{1}-norm in terms of the L_{2}-norm and the eigenvalue counting function. This estimate is applied to a comparison of the heat content with the heat trace of the semigroup.

MSC 2010: 35P99, 35K08, 35J10, 47A10 Keywords: Schrödinger operators, eigenfunctions, L_{1}-estimates,

heat kernel estimates, heat content, heat trace

1 Introduction and main results

In the recent paper [BHV13], the authors studied Dirichlet Laplacians on open subsets Ω of ℝ^{d}. They proved an estimate for the L_{1}-norm of eigenfunctions in terms of their L_{2}-norm and spectral data, and they used this to estimate the heat content of Ω by its heat trace. The aim of the present paper is to provide sharper estimates in the following more general setting.

Let Ω ⊆ ℝ^{d} be measurable, where d ∈ ℕ, and let T be a selfadjoint positivity preserving C_{0}-semigroup on L_{2}(Ω) that is dominated by the free heat semigroup, i.e.,

0 ⩽ T (t)f ⩽ e^{tΔ}f (t ⩾ 0, 0 ⩽ f ∈ L_{2}(Ω)).

Let −H denote the generator of T.

∗Fachbereich 3 – Mathematik, Universität Bremen, 28359 Bremen, Germany, +49 421 218-63702, hendrik.vohugo@egon gt@uni-hannover bremen.de

An important example for the operator −H is the Dirichlet Laplacian with a locally integrable absorption potential on an open set Ω ⊆ ℝ^{d}. For more general absorption potentials the space of strong continuity of the semigroup will be L_{2}(Ω′) for some measurable Ω′ ⊆ Ω.

In our first main result we estimate the L_{1}-norm of eigenfunctions of H in terms of their L_{2}-norm and the eigenvalue counting function N_{t}(H), which for t < inf σ_{ess}(H) denotes the number of eigenvalues of H that are ⩽ t, counted with multiplicity.

1.1 Theorem. Let φ be an eigenfunction of H with eigenvalue λ < inf σ_{ess}(H). Then

||φ||_{1}^{2} ⩽ c_{d}(t − λ)^{−d/2}(ln \frac{3t}{t−λ})^{d}N_{t}(H)||φ||_{2}^{2} (λ < t < inf σ_{ess}(H)),

with c_{d} = 35^{d+1}d^{d/2}.

1.2 Remarks. (a) We point out that as in [BHV13; Thm. 1.6] one has the lower bound

||φ||_{1}^{2} ⩾ (\frac{2πd}{e})^{d/2}λ^{−d/2}||φ||_{2}^{2} .

Thus, the factor d^{d/2} in the constant c_{d} is of the correct order. The factor (t−λ)^{−d/2} matches the factor λ^{−d/2}; cf. Corollary 1.3 below. See [BHV13; Example 1.8(3)] for an explanation why the estimate should contain a factor like N_{t}(H) with some t > λ.

(b) In [BHV13; Thm. 1.3], in the framework of Dirichlet Laplacians on open subsets of ℝ^{d}, the estimate

||φ||_{1}^{2} ⩽ C_{d}E_{0}^{−d/2} ((\frac{λ}{E_{0}} \frac{λ}{t − λ})^{d}(ln N_{t}(H))^{d}N_{t}(H) + (\frac{λ}{E_{0}} )^{4d−3}(\frac{λ}{t − λ})^{4d} ) ||φ||_{2}^{2}

was shown under the additional assumption t ⩽ 3λ, where E_{0} = inf σ(H). Our estimate ||φ||_{1}^{2} ⩽ c_{d}λ^{−d/2}(\frac{λ}{t−λ})^{d/2}(ln \frac{3t}{t−λ})^{d}N_{t}(H)||φ||_{2}^{2} improves on this in several regards; most notably, the factors \frac{λ}{E_{0}} , (ln N_{t}(H))^{d} and the second summand are removed altogether.

(c) In [BHV13], a partition of ℝ^{d} into cubes was used in the proof. We will work with a “continuous partition” into balls instead; see the proof of Lemma 4.2. Working with balls leads to a better constant c_{d} in the estimate.

(d) In the case d = 1 and H the Dirichlet Laplacian on an open subset of ℝ, an improved estimate is given in [BHV13; Rem. 1.5]. For that estimate it is crucial that H is a direct sum of Dirichlet Laplacians on intervals. The improvement is not possible for general H in dimension d = 1; this can be seen similarly as in [BHV13; Example 1.8(3)].

If H has compact resolvent, then one can apply Theorem 1.1 with t = (1 + ε)λ for any ε > 0 to obtain the following estimate. Note that it contains the same factor λ^{−d/2} as the lower bound of Remark 1.2(a).

1.3 Corollary. Assume that H has compact resolvent. Let φ be an eigenfunction of H with eigenvalue λ. Then

||φ||_{1}^{2} ⩽ c_{d}C_{ε}^{d}λ^{−d/2}N_{(1+ε)λ}(H)||φ||_{2}^{2} (ε > 0),

with c_{d} as in Theorem 1.1 and C_{ε} = ε^{−1/2} ln(3 + \frac{3}{ε}).

1.4 Remark. The assumption that H has compact resolvent is in particular satisfied if Ω has finite volume. Note that then the trivial estimate ||φ||_{1}^{2} ⩽ vol(Ω)||φ||_{2}^{2} holds. We point out that, up to a dimension dependent constant, the estimate of Corollary 1.3 is never worse since one has the bound N_{t}(H) ⩽ K_{d} vol(Ω)t^{d/2} for all t > 0. (To obtain this bound, apply [LiYa83; Cor. 1] to open sets Ω̃ ⊇ Ω and note that e^{−tH} ⩽ e^{−tΔΩ̃}, where ΔΩ̃ denotes the Dirichlet Laplacian on Ω̃.)

Our second main result is the following heat kernel estimate for semigroups dominated by the free heat semigroup. This estimate is obtained as a by-product of the preparations for the proof of Theorem 1.1.

1.5 Theorem. For all t > 0 the semigroup operator e^{−tH} has an integral kernel p_{t}. If E_{0} := inf σ(H) > 0 then

0 ⩽ p_{t}(x, y) ⩽ (\frac{eE_{0}}{2πd})^{d/2} exp (−E_{0}t − \frac{|x − y|^{2}}{4t} ) (t ⩾ \frac{d}{2E_{0}} , x, y ∈ ℝ^{d}).

1.6 Remark. (a) For 0 < t < \frac{d}{2E_{0}} one just has the estimate with respect to the free heat kernel,

0 ⩽ p_{t}(x, y) ⩽ (4πt)^{−d/2}exp (−\frac{|x − y|^{2}}{4t} ).

In combination with Theorem 1.5 this gives

0 ⩽ p_{t}(x, y) ⩽ (4πt)^{−d/2} (1 + \frac{2e}{d}E_{0}t)^{d/2} exp (−E_{0}t − \frac{|x − y|^{2}}{4t} ) (t > 0).

(In the case E_{0} = 0 this estimate is true but inconsequential.)

(b) In [Ouh06; formula (22)], the following estimate was proved in the framework of Dirichlet Laplacians with absorption potentials on open subsets of ℝ^{d}:

p_{t}(x, y) ⩽ c_{ε}(4πt)^{−d/2} (1 + \frac{1}{2}E_{0}t + ε\frac{|x − y|^{2}}{8t} )^{d/2} exp (−E_{0}t − \frac{|x − y|^{2}}{4t} ) ,

where ε > 0 and c_{ε} = e^{2}(1 + \frac{1}{ε})^{d/2}. Part (a) shows that the summand ε\frac{|x−y|^{2}}{8t} is actually not needed, which may come as a surprise.

(c) In the generality of our setting, the estimate provided in Theorem 1.5 is probably the best one can hope for. Suppose, for example, that the semigroup T is irreducible and that E_{0} is an isolated eigenvalue of H. Then the large time behaviour of p_{t} is known:

e^{E_{0}t}p_{t}(x, y) → φ(x)φ(y) (t → ∞),

where φ is the non-negative normalized ground state of H; see, e.g., [KLVW13; Thm. 3.1]. Moreover, if inf σ(H) = 1 then E^{d/4}φ(E^{1/2}·) is the ground state of an appropriately scaled operator H_{E} with inf σ(H_{E}) = E. This explains the factor E_{0}^{d/2} in our estimate.

Note, however, that better estimates are known for Dirichlet Laplacians under suitable geometric assumptions on the domain Ω. Then a boundary term like φ(x)φ(y) can be included in the estimate. This can be shown via intrinsic ultracontractivity as in [OuWa07].

An important application of Corollary 1.3 is that it allows us to compare the “heat content” of H with its “heat trace”. We assume that H has compact resolvent, with (λ_{k}) the increasing sequence of all the eigenvalues of H, repeated according to their multiplicity. For t > 0 we denote by Q_{H}(t) := ||e^{−tH}𝟙_{Ω}||_{1} the heat content, by Z_{H}(t) := ∑_{k=1}^{∞} e^{−tλ_{k}} the heat trace of H.

Note that Q_{H}, Z_{H} are decreasing functions. It may well occur that Q_{H}(t) = ∞ and/or Z_{H}(t) = ∞ for some but not all t > 0 if Ω has infinite Lebesgue measure, see [BeDa89; Thm. 5.5].

1.7 Theorem. Assume that H has compact resolvent and that Z_{H}(t_{0}) < ∞ for some t_{0} > 0. Then Q_{H}(t) < ∞ for all t > 2t_{0},

Q_{H}(t) ⩽ c_{ε,d}λ_{1}^{−d/2}Z_{H}(\frac{t}{2+ε})^{2} (0 < ε < \frac{t}{t_{0}} − 2),

with c_{ε,d} = c_{d}C_{ε}^{d} as in Corollary 1.3.

The proof is rather short, so we give it right here. We will use the following simple estimate.

1.8 Lemma. (cf. [BHV13; Lemma 5.2]) For T, λ > 0 one has N_{λ}(H) ⩽ Z_{H}(T)e^{T λ}.

Proof. If k ∈ ℕ is such that λ_{k} ⩽ λ, then k ⩽ e^{T λ_{k}} ∑_{j=1}^{k} e^{−T λ_{j}} ⩽ e^{T λ}Z_{H}(T). Thus, N_{λ}(H) = #{k; λ_{k} ⩽ λ} ⩽ e^{T λ}Z_{H}(T).

Proof of Theorem 1.7. Let T := \frac{t}{2+ε}. Let (φ_{k}) be an orthonormal basis of L_{2}(Ω) such that Hφ_{k} = λ_{k}φ_{k} for all k ∈ ℕ. By Corollary 1.3 and Lemma 1.8 we obtain

||φ_{k}||_{1}^{2} ⩽ c_{ε,d}λ_{k}^{−d/2}N_{(1+ε)λ_{k}}(H)||φ_{k}||_{2}^{2} ⩽ c_{ε,d}λ_{1}^{−d/2}Z_{H}(T)e^{T (1+ε)λ_{k}}

for all k ∈ ℕ. For f ∈ L_{2}(Ω) ∩ L_{∞}(Ω) one has e^{−tH}f = ∑_{k=1}^{∞}⟨f, φ_{k}⟩e^{−tλ_{k}}φ_{k} and hence

||e^{−tH}f||_{1} ⩽ ∑_{k=1}^{∞} ||f||_{∞}e^{−tλ_{k}}||φ_{k}||_{1}^{2} .

Using a sequence (f_{k}) in L_{2}(Ω) with 0 ⩽ f_{k} ↑ 𝟙_{Ω} and recalling T (1 + ε) = t − T, we conclude that

||e^{−tH}𝟙_{Ω}||_{1} ⩽ ∑_{k=1}^{∞} e^{−tλ_{k}}||φ_{k}||_{1}^{2} ⩽ c_{ε,d}λ_{1}^{−d/2}Z_{H}(T) ∑_{k=1}^{∞} e^{−T λ_{k}} = c_{ε,d}λ_{1}^{−d/2}Z_{H}(T)^{2}.

The paper is organized as follows. In Section 2 we investigate properties of selfadjoint positivity preserving semigroups dominated by the free heat semigroup. In Section 3 we prove Theorem 1.5, and we show off-diagonal resolvent estimates needed in the proof of Theorem 1.1, which in turn is given in Section 4.

2 Semigroups dominated by the free heat semigroup

Throughout this section let Ω ⊆ ℝ^{d} be measurable, and let T be a selfadjoint positivity preserving C_{0}-semigroup on L_{2}(Ω) that is dominated by the free heat semigroup, with generator −H. Let τ be the closed symmetric form associated with H. The purpose of this section is to collect some basic properties of τ and H.

It is crucial that D(τ) is a subset of H^{1}(ℝ^{d}) (in fact an ideal; see, e.g., [MVV05; Cor. 4.3]). Thus we can define a symmetric form σ by

σ(u, v) := τ(u, v) − ⟨∇u, ∇v⟩ (u, v ∈ D(σ) := D(τ)).  (2.1)

This gives a decomposition of the form τ as the standard Dirichlet form plus a form σ that is positive and local in the sense of the following lemma. If −H is the Dirichlet Laplacian with an absorption potential V ⩾ 0 on an open set Ω ⊆ ℝ^{d}, then σ(u, v) = ∫ V u\overline{v}. In this case the next three results are trivial.

2.1 Lemma. Let 0 ⩽ u, v ∈ D(τ). Then σ(u, v) ⩾ 0, and σ(u, v) = 0 if u ∧ v = 0.

Proof. By [MVV05; Cor. 4.3], the first assertion follows from the assumption that T is a positive semigroup dominated by the free heat semigroup. For the second assertion let w := u − v. Then τ(u, v) = τ(w^{+}, w^{−}) ⩽ 0 since T is a positive semigroup (see, e.g., [MVV05; Cor. 2.6]). Since ⟨∇u, ∇v⟩ = 0, this implies σ(u, v) ⩽ 0 and hence σ(u, v) = 0.

2.2 Lemma. If ξ ∈ W_{∞}^{1}(ℝ^{d}) and u ∈ D(τ), then ξu ∈ D(τ). Moreover, f: ℝ^{d} → D(τ), f(x) := ξ(· − x)u is continuous.

Proof. By [MVV05; Cor. 4.3], D(τ) is an ideal of H^{1}(ℝ^{d}). This implies the first assertion ξu ∈ D(τ) since ξu ∈ H^{1}(ℝ^{d}) and |ξu| ⩽ ||ξ||_{∞}|u| ∈ D(τ).

For the second assertion it suffices to show continuity at 0, and we can assume without loss of generality that ξ, u are real-valued. From the identity

f(x) − f(0) = ξ(· − x)(u − u(· − x)) + (ξu)(· − x) − ξu

one deduces that f: ℝ^{d} → H^{1}(ℝ^{d}) is continuous at 0. By Lemma 2.1 we obtain

σ(f(x) − f(0)) = σ(|f(x) − f(0)|) ⩽ σ(||ξ(· − x) − ξ||_{∞}|u|) ⩽ ||∇ξ||_{∞}^{2}|x|^{2}σ(|u|).

Due to the decomposition (2.1) this yields continuity of f: ℝ^{d} → D(τ) at 0.

2.3 Lemma. Let u, v ∈ D(τ). Then σ(ξu, v) = σ(u, ξv) for all ξ ∈ W_{∞}^{1}(ℝ^{d}).

Proof. Since D(τ) is a lattice, it suffices to show the assertion for u, v ⩾ 0 and real-valued ξ. Throughout the proof we consider only real-valued function spaces. We define a bilinear form b by

b(φ, ψ) := σ(φu, ψv) (φ, ψ ∈ D(b) := W_{∞,0}^{1}(ℝ^{d})).

Then b(φ, ψ) ⩾ 0 for φ, ψ ⩾ 0 by Lemma 2.1. Now one can proceed similarly as in [ArWa03; proof of Thm. 4.1] to show that

σ(φu, ψv) = ∫ φψ dμ (φ, ψ ∈ W_{∞,0}^{1}(ℝ^{d}))  (2.2)

for some finite positive Borel measure μ on ℝ^{d} (depending of course on u, v). We only sketch the argument: first one can extend b to a continuous bilinear form on C_{0}(ℝ^{d}), by positivity. Then one uses the linearisation of b in C_{0}(ℝ^{d} × ℝ^{d})′ to obtain a finite Borel measure ν on ℝ^{d} × ℝ^{d} such that b(φ, ψ) = ∫ φ(x)ψ(y) dν(x, y) for all φ, ψ ∈ W_{∞,0}^{1}(ℝ^{d}). Finally, spt ν ⊆ {(x, x); x ∈ ℝ^{d}} since b(φ, ψ) = 0 in the case spt φ ∩ spt ψ = ∅, by Lemma 2.1, and this leads to the asserted measure μ.

To complete the proof, we show that the representation (2.2) is valid for all φ, ψ ∈ W_{∞}^{1}(ℝ^{d}). Let χ ∈ C_{c}^{1}(ℝ^{d}) such that 0 ⩽ χ ⩽ 1 and χ|_{B(0,1)} = 1. Then u_{n} := χ(\frac{·}{n})u → u in H^{1}(ℝ^{d}) as n → ∞, and σ(u_{n}) ⩽ σ(u) for all n ∈ ℕ by Lemma 2.1. Therefore, lim sup τ(u_{n}) ⩽ τ(u), and this implies u_{n} → u in D(τ). Applying (2.2) to σ(χ(\frac{·}{n})φu, χ(\frac{·}{n})ψv) and letting n → ∞ we derive (2.2) for any φ, ψ ∈ W_{∞}^{1}(ℝ^{d}). For real-valued ξ ∈ W_{∞}^{1}(ℝ^{d}) we now obtain

σ(ξu, v) = ∫ ξ dμ = σ(u, ξv).

In the proof of Theorem 1.1 we will work with operators that are subordinated to H as follows. For an open set U ⊆ ℝ^{d} let H_{U} denote the selfadjoint operator in L_{2}(Ω ∩ U) associated with the form τ restricted to D(τ) ∩ H_{0}^{1}(U). (Observe that this form domain is dense in L_{2}(Ω ∩ U).)

2.4 Lemma. Let φ be an eigenfunction of H with eigenvalue λ. Let U be an open subset of ℝ^{d}, and let ξ ∈ W_{∞}^{2}(ℝ^{d}), ξ = 0 on ℝ^{d} \ U. Then ξφ ∈ D(H_{U}) and

(H_{U} − λ)(ξφ) = −2∇ξ · ∇φ − (Δξ)φ.

Proof. By Lemma 2.2 we have ξφ ∈ D(τ). Moreover, ξφ ∈ H_{0}^{1}(U) due to the assumption ξ = 0 on ℝ^{d} \ U. For v ∈ D(τ) ∩ H_{0}^{1}(U) we have ξv ∈ D(τ) ∩ H_{0}^{1}(U) and

(τ − λ)(φ, ξv) = ⟨(H − λ)φ, ξv⟩ = 0.

Since σ(ξφ, v) = σ(φ, ξv) by Lemma 2.3, the decomposition (2.1) yields

(τ − λ)(ξφ, v) = (τ − λ)(ξφ, v) − (τ − λ)(φ, ξv)
= ⟨∇(ξφ), ∇v⟩ − ⟨∇φ, ∇(ξv)⟩ = ⟨φ∇ξ, ∇v⟩ − ⟨∇φ, v∇ξ⟩.

Now φ∇ξ is in H^{1}(ℝ^{d})^{d} and ∇ · (φ∇ξ) = ∇φ · ∇ξ + φΔξ, so we conclude that

(τ − λ)(ξφ, v) = −⟨2∇φ · ∇ξ + φΔξ, v⟩

for all v ∈ D(τ) ∩ H_{0}^{1}(U), which proves the assertion.

3 Heat kernel estimates

In this section we prove Theorem 1.5, and we provide resolvent estimates needed in the proof of Theorem 1.1. Throughout we denote

ℂ_{+} := {z ∈ ℂ; Re z > 0}.

We point out that in the following result T is not required to be a semigroup.

3.1 Proposition. Let (Ω, μ) be a measure space, and let ρ: Ω → ℝ be measurable. Let λ ∈ ℝ, and let T: ℂ_{+} → L(L_{2}(μ)) be analytic, ||T (z)|| ⩽ e^{−λ Re z} for all z ∈ ℂ_{+}. Assume that there exists C ⩾ 0 such that

||e^{αρ}T (t)e^{−αρ}|| ⩽ Ce^{α^{2}t} (α, t > 0).

Then

||e^{αρ}T (z)e^{−αρ}|| ⩽ exp(α^{2}/ Re \frac{1}{z} − λ Re z) (α > 0, z ∈ ℂ_{+}),

in particular, ||e^{αρ}T (t)e^{−αρ}|| ⩽ e^{α^{2}t−λt} for all α, t > 0.

Here and in the following we denote

||wBw^{−1}|| := sup{||wBw^{−1}f||_{2}; f ∈ L_{2}(μ), ||f||_{2} ⩽ 1, w^{−1}f ∈ L_{2}(μ)}

for an operator B ∈ L(L_{2}(μ)) and a measurable function w: Ω → (0, ∞).

Proof of Proposition 3.1. Observe that

M := {f ∈ L_{2}(μ); ρ bounded on [f ≠ 0]}

is dense in L_{2}(μ). Let α > 0, and let f, g ∈ M with ||f||_{2} = ||g||_{2} = 1. Define the analytic function F: ℂ_{+} → ℂ by

F (z) := e^{λz−α^{2}/z}⟨e^{αρ/z}T (z)e^{−αρ/z}f, g⟩.

Let c > 0 such that |ρ| ⩽ c on [f ≠ 0] ∪ [g ≠ 0]. Then

|F (z)| ⩽ ||e^{λz}T (z)||||e^{−αρ/z}f||_{2}||e^{αρ/z̄}g||_{2} ⩽ exp(2αc Re \frac{1}{z}) (z ∈ ℂ_{+}),

in particular |F (t)| ⩽ e^{2αc} for all t ⩾ 1. Moreover,

|F (t)| ⩽ e^{λt−α^{2}/t}||e^{αρ/t}T (t)e^{−αρ/t}|| ⩽ e^{λt−α^{2}/t} · Ce^{(α/t)^{2}t} ⩽ Ce^{|λ|}

for all 0 < t < 1. Thus, |F (z)| ⩽ 1 for all z ∈ ℂ_{+} by the next lemma, and this yields

||e^{αρ/z}T (z)e^{−αρ/z}|| ⩽ exp(α^{2} Re \frac{1}{z} − λ Re z) (α > 0, z ∈ ℂ_{+}).

The assertion follows by replacing α with α/ Re \frac{1}{z}.

The following Phragmén-Lindelöf type result is similar to [CoSi08; Prop. 2.2].

3.2 Lemma. Let F: ℂ_{+} → ℂ be analytic. Assume that there exist c_{1}, c_{2} > 0 such that

|F (z)| ⩽ exp(c_{1} Re \frac{1}{z}) (z ∈ ℂ_{+}), |F (t)| ⩽ c_{2} (t > 0).

Then |F (z)| ⩽ 1 for all z ∈ ℂ_{+}.

Proof. Note that lim sup_{z→iy} |F (z)| ⩽ 1 for all y ∈ ℝ \ {0}. Thus, |F (z)| ⩽ c_{2} ∨ 1 for all z ∈ ℂ_{+} by the Phragmén-Lindelöf principle applied to the sectors {z ∈ ℂ; Re z > 0, Im z > 0} and {z ∈ ℂ; Re z > 0, Im z < 0}. Then an application of the Phragmén-Lindelöf principle to the sector ℂ_{+} implies |F | ⩽ 1 on ℂ_{+}.

In the next lemma we state a version of the well-known Davies’ trick; cf. [Dav95; proof of Lemma 19]. For the proof note that inf_{ξ∈ℝ^{d}} exp(|ξ|^{2}t − ξ · x) = exp(−\frac{|x|^{2}}{4t} ) for all t > 0, x ∈ ℝ^{d}.

3.3 Lemma. Let Ω ⊆ ℝ^{d} be measurable, and let B be a positive operator on L_{2}(Ω). For ξ, x ∈ ℝ^{d} let ρ_{ξ}(x) := e^{ξx}. Then for t > 0 the following are equivalent:

(i) B ⩽ e^{tΔ}, (ii) ||ρ_{ξ}Bρ_{ξ}^{−1}||_{1→∞} ⩽ (4πt)^{−d/2}e^{|ξ|^{2}t} for all ξ ∈ ℝ^{d}.

In (i), the inequality B ⩽ e^{tΔ} is meant in the sense of positivity preserving operators, i.e.,

Bf ⩽ e^{tΔ}f (0 ⩽ f ∈ L_{2}(Ω)).

The following result provides an estimate of the resolvent of H by the free resolvent. Together with Proposition 3.5 below this will be an important stepping stone in the proof of Theorem 1.1.

3.4 Theorem. Let Ω ⊆ ℝ^{d} be measurable, and let T be a selfadjoint positive C_{0}- semigroup on L_{2}(Ω) that is dominated by the free heat semigroup. Let −H be the generator of T, and let E_{0} := inf σ(H). Then for all ε ∈ (0, 1] one has

T (t) ⩽ ε^{−d/2}e^{−(1−ε)E_{0}t}e^{tΔ}  (t ⩾ 0),
(H − λ)^{−1} ⩽ ε^{−d/2}((1 − ε)E_{0} − λ − Δ)^{−1}  (λ < (1 − ε)E_{0}).

Proof. As above let ρ_{ξ}(x) := e^{ξx}. The assumptions imply ||T (z)||_{2→2} ⩽ e^{−E_{0} Re z} for all z ∈ ℂ_{+} and

||ρ_{ξ}T (t)ρ_{ξ}^{−1}||_{2→2} ⩽ ||ρ_{ξ}e^{tΔ}ρ_{ξ}^{−1}||_{2→2} = e^{t|ξ|^{2}} (ξ ∈ ℝ^{d}, t ⩾ 0).

By Proposition 3.1 it follows that

||ρ_{ξ}T (t)ρ_{ξ}^{−1}||_{2→2} ⩽ e^{t|ξ|^{2}−E_{0}t} (ξ ∈ ℝ^{d}, t ⩾ 0).  (3.1)

Let t > 0, and let k_{t} be the convolution kernel of e^{tΔ}. Then for ξ ∈ ℝ^{d} the kernel of e^{−t|ξ|^{2}}ρ_{ξ}e^{tΔ}ρ_{ξ}^{−1} is given by

e^{−t|ξ|^{2}+ξ·(x−y)}k_{t}(x − y) = k_{t}(x − 2tξ − y) (x, y ∈ ℝ^{d})

since −t|ξ|^{2} + ξ · (x − y) − \frac{|x−y|^{2}}{4t} = −\frac{|x−y−2tξ|^{2}}{4t}. (The above identity is the key point in the proof; this is why we need unbounded weights in Proposition 3.1.) Therefore,

e^{−t|ξ|^{2}}||ρ_{ξ}T (t)ρ_{ξ}^{−1}||_{2→∞} ⩽ ||e^{tΔ}||_{2→∞} = ||e^{2tΔ}||_{1→∞}^{1/2} = (8πt)^{−d/4}.

By duality we also have e^{−t|ξ|^{2}}||ρ_{ξ}T (t)ρ_{ξ}^{−1}||_{1→2} ⩽ (8πt)^{−d/4}. Using the semigroup property and (3.1), we conclude for ε ∈ (0, 1] that

||ρ_{ξ}T (t)ρ_{ξ}^{−1}||_{1→∞} ⩽ ||ρ_{ξ}T (\frac{ε}{2}t)ρ_{ξ}^{−1}||_{2→∞}||ρ_{ξ}T ((1 − ε)t)ρ_{ξ}^{−1}||_{2→2}||ρ_{ξ}T (\frac{ε}{2}t)ρ_{ξ}^{−1}||_{1→2}

⩽ e^{t|ξ|^{2}}(8π \frac{ε}{2}t)^{−d/4}e^{−E_{0}(1−ε)t}(8π \frac{ε}{2}t)^{−d/4} = (4πεt)^{−d/2}e^{t|ξ|^{2}−E_{0}(1−ε)t}.

Now the first assertion follows from Lemma 3.3, and this gives the second assertion by the resolvent formula.

Proof of Theorem 1.5. The existence of the kernel p_{t} follows from the Dunford-Pettis theorem, and Theorem 3.4 implies

p_{t}(x, y) ⩽ (4πεt)^{−d/2}e^{εE_{0}t} exp (−E_{0}t − \frac{|x − y|^{2}}{4t} )

for all t > 0. Then for t ⩾ \frac{d}{2E_{0}} the assertion follows by setting ε := \frac{d}{2E_{0}t}.

We conclude this section with an off-diagonal L_{1}-estimate for the free resolvent.

3.5 Proposition. Let A, B ⊆ ℝ^{d} be measurable, and let d(A, B) denote the distance between A and B. Then

||𝟙_{A}(μ − Δ)^{−1}𝟙_{B}||_{1→1} ⩽ (1 − θ^{2})^{−d/2} \frac{1}{μ} exp(−θ\sqrt{μ} d(A, B))

for all μ > 0, 0 < θ < 1.

Proof. Let r := d(A, B). By duality we have to show

||𝟙_{B}(μ − Δ)^{−1}𝟙_{A}||_{∞→∞} ⩽ (1 − θ^{2})^{−d/2} \frac{1}{μ} exp(−θr\sqrt{μ}) =: C,

or equivalently, (μ − Δ)^{−1}𝟙_{A} ⩽ C on B. Let x ∈ B. By the resolvent formula we obtain

(μ − Δ)^{−1}𝟙_{A}(x) = ∫_{0}^{∞}e^{−μt}∫_{ℝ^{d}} k_{t}(y)𝟙_{A}(x − y) dy dt ⩽ ∫_{0}^{∞}e^{−μt}∫_{|y|⩾r} k_{t}(y) dy dt,

where k_{t}(y) = (4πt)^{−d/2} exp(−\frac{|y|^{2}}{4t} ). We substitute y = (4t)^{1/2}z and note that |y| ⩾ r if and only if t ⩾ (\frac{r}{2|z|})^{2}; then by Fubini’s theorem we infer that

(μ − Δ)^{−1}𝟙_{A}(x) ⩽ π^{−d/2} ∫_{ℝ^{d}} ∫(\frac{r}{2|z|})^{2}^{∞} e^{−μt} dt e^{−|z|^{2}} dz

= π^{−d/2} \frac{1}{μ} ∫_{ℝ^{d}} exp(−\frac{μr^{2}}{4|z|^{2}} − |z|^{2}) dz.

Note that θr\sqrt{μ} ⩽ \frac{μr^{2}}{4|z|^{2}} + θ^{2}|z|^{2} and hence exp(−\frac{μr^{2}}{4|z|^{2}} − |z|^{2}) ⩽ e^{−θr\sqrt{μ}}e^{−(1−θ^{2})|z|^{2}} for all z ∈ ℝ^{d}. We conclude that

(μ − Δ)^{−1}𝟙_{A}(x) ⩽ \frac{1}{μ}e^{−θr\sqrt{μ}} π^{−d/2}∫_{ℝ^{d}} e^{−(1−θ^{2})|z|^{2}}dz = \frac{1}{μ}e^{−θr\sqrt{μ}}(1 − θ^{2})^{−d/2},

which proves the assertion.

3.6 Remark. For μ > (\frac{d}{r})^{2} (where r = d(A, B)), optimizing the estimate of Proposition 3.5 with respect to θ leads to the choice θ = (1− \frac{d}{r\sqrt{μ}})^{1/2}. For μ > (\frac{d}{2r})^{2}, the choice θ = 1 − \frac{d}{2r\sqrt{μ}} yields

||𝟙_{A}(μ − Δ)^{−1}𝟙_{B}||_{1→1} ⩽ (\frac{2e}{d} + r\sqrt{μ})^{d/2} \frac{1}{μ}e^{−r\sqrt{μ}}.

4 Proof of Theorem 1.1

Throughout this section we assume the setting of Section 2, i.e., Ω ⊆ ℝ^{d} is measurable, T a selfadjoint positivity preserving C_{0}-semigroup on L_{2}(Ω) dominated by the free heat semigroup, with generator −H, and τ the closed symmetric form associated with H. We denote

E_{0}(H) := inf σ(H).

Recall that, for an open set U ⊆ ℝ^{d}, H_{U} is the selfadjoint operator in L_{2}(Ω ∩ U) associated with the form τ restricted to D(τ) ∩ H_{0}^{1}(U).

For A ⊆ ℝ^{d} we denote by U_{ε}(A) = ⋃_{x∈A} B(x, ε) the ε-neighborhood of A. If A is measurable, then we write |A| for the Lebesgue measure of A. For r > 0 and E_{0}(H) < t < inf σ_{ess}(H) we define the sets

\begin{aligned}F_{r}(t) & := \{x ∈ ℝ^{d}; E_{0}(H_{B(x,r)}) < t\}, \\ G_{r}(t) & := ℝ^{d} \ \overline{U_{r}(F_{r}(t))}. \end{aligned}  (4.1)

For the proof of Theorem 1.1 the following two facts will be crucial. On the one hand, the set F_{r}(t) is “small” in the sense that the Lebesgue measure of U_{3r}(F_{r}(t)) is not too large, as is expressed in the next lemma. On the other hand, the set G_{r}(t) is “spectrally small” in the sense that the ground state energy of H_{G_{r}(t)} is not much smaller than t; see Lemma 4.2 below.

4.1 Lemma. Let r > 0 and E_{0}(H) < t < inf σ_{ess}(H). Then

|U_{s}(F_{r}(t))| ⩽ ω_{d}(2r + s)^{d}N_{t}(H) (s > 0),

where ω_{d} := |B(0, 1)|.

Proof. Let M ⊆ F_{r}(t) be a maximal subset with the property that the balls B(x, r), x ∈ M are pairwise disjoint. Then by the min-max principle and the definition of F_{r}(t) one sees that M has at most N_{t}(H) elements. Moreover, F_{r}(t) ⊆ ⋃_{x∈M} B(x, 2r) by the maximality of M. Therefore,

|U_{s}(F_{r}(t))| ⩽ ∑_{x∈M} |B(x, 2r + s)| ⩽ N_{t}(H) · ω_{d}(2r + s)^{d}.

4.2 Lemma. Let E_{0,d} denote the ground state energy of the Dirichlet Laplacian on B(0, 1). Then E_{0,d} ⩽ \frac{1}{2}(d + 1)(d + 2) ⩽ \frac{3}{4}(d + 1)^{2}, and

E_{0}(H_{G_{r}(t)}) ⩾ t − E_{0,d}/r^{2} (r > 0, E_{0}(H) < t < inf σ_{ess}(H)).

Proof. For ψ ∈ W_{2,0}^{1}(B(0, 1)) defined by ψ(x) = 1 − |x| one easily computes ||∇ψ||_{2}^{2}/||ψ||_{2}^{2} = \frac{1}{2}(d + 1)(d + 2), thus proving the first assertion. Let now ψ denote the normalized ground state of the Dirichlet Laplacian on B(0, 1). For r > 0 let ψ_{r} := r^{−d/2}ψ(\frac{·}{r}); note that ||ψ_{r}||_{2} = 1 and ψ_{r} ∈ W_{∞}^{1}(ℝ^{d}).

To prove the second assertion, we need to show that

τ(u) ⩾ (t − E_{0,d}/r^{2})||u||_{2}^{2}  (4.2)

for all u ∈ D(τ) ∩ H_{0}^{1}(G_{r}(t)), without loss of generality u real-valued. We will use (ψ_{r}(· − x)^{2})_{x∈ℝ^{d}} as a continuous partition of the identity. By Lemma 2.2 we have ψ_{r}(· − x)u ∈ D(τ) for all x ∈ ℝ^{d}. Using (2.1) and Lemma 2.3 we obtain

τ(ψ_{r}(· − x)u) = ||ψ_{r}(· − x)∇u + u∇ψ_{r}(· − x)||_{2}^{2} + σ(ψ_{r}(· − x)u)

= ∫ (∇(ψ_{r}(· − x)^{2}u) · ∇u + u^{2}|∇ψ_{r}(· − x)|^{2}) + σ(ψ_{r}(· − x)^{2}u, u)

= τ(ψ_{r}(· − x)^{2}u, u) + ∫ u^{2}|∇ψ_{r}(· − x)|^{2}.

Note that ∫ ψ_{r}(y − x)^{2} dx = ||ψ_{r}||_{2}^{2} = 1 and

∫ |∇ψ_{r}(y − x)|^{2} dx = ||∇ψ_{r}||_{2}^{2} = ||∇ψ||_{2}^{2}/r^{2} = E_{0,d}/r^{2}

for all y ∈ ℝ^{d}. Taking into account Lemma 2.2 (with ξ = ψ_{r}^{2}) we thus obtain ∫ τ(ψ_{r}(· − x)^{2}u, u) dx = τ(u, u) and hence

∫ τ(ψ_{r}(· − x)u) dx = τ(u) + ||u||_{2}^{2} · E_{0,d}/r^{2}.

To conclude the proof of (4.2), we show that the left hand side of this identity is greater or equal t||u||_{2}^{2}: note that ψ_{r}(· − x)u ∈ H_{0}^{1}(B(x, r)). For x ∈ ℝ^{d} \ F_{r}(t) we have τ(ψ_{r}(· − x)u) ⩾ t||ψ_{r}(· − x)u||_{2}^{2} by the definition of F_{r}(t); for x ∈ F_{r}(t) we have ψ_{r}(· − x)u = 0 since u ∈ H_{0}^{1}(G_{r}(t)). Therefore,

∫ τ(ψ_{r}(· − x)u) dx ⩾ t ∫ ||ψ_{r}(· − x)u||_{2}^{2} dx = t||u||_{2}^{2} .

4.3 Remark. It is known that E_{0,d} behaves like \frac{1}{4}d^{2} for large d. For d = 3, however, the estimate E_{0,d} ⩽ \frac{1}{2}(d + 1)(d + 2) = 10 from Lemma 4.2 is quite sharp since E_{0,3} = π^{2} > 9.86.

4.4 Lemma. There exists 0 ⩽ ρ ∈ C^{2}(ℝ^{d}) such that spt ρ ⊆ B(0, 1), ∫ ρ = 1 and

||∇ρ||_{1} ⩽ d + 1,  ||Δρ||_{1} ⩽ 2(d + 1)^{2}.  (4.3)

Proof. Let ρ_{0} ∈ W_{1}^{1}(ℝ^{d}), ρ_{0}(x) := \frac{d(d+2)}{2σ_{d−1}}(1 − |x|^{2})𝟙_{B(0,1)}(x), where σ_{d−1} denotes the surface measure of the unit sphere ∂B(0, 1). Then one easily computes

∫ ρ_{0} = 1, ||∇ρ_{0}||_{1} = \frac{d(d + 2)}{d + 1} < d + 1

and

Δρ_{0} = \frac{d(d + 2)}{σ_{d−1}}(−d𝟙_{B(0,1)} + δ_{∂B(0,1)})

in the distributional sense, so Δρ_{0} is a measure with ||Δρ_{0}|| = 2d(d+2) < 2(d+1)^{2}. Using a suitable mollifier and scaling, one obtains ρ as asserted.

Proof of Theorem 1.1. (i) Let r > (\frac{E_{0,d}}{t−λ} )^{1/2}, and let F_{r} := F_{r}(t), G_{r} := G_{r}(t) be as in (4.1). Then E_{0}(H_{G_{r}}) > λ by Lemma 4.2. We define ξ ∈ C^{2}(ℝ^{d}) satisfying

spt ξ ⊆ G_{r}, spt(𝟙_{ℝ^{d}} − ξ) ⊆ U_{2r}(F_{r})

as follows: let ρ_{r} := r^{−d}ρ(\frac{·}{r}), where ρ is as in Lemma 4.4. Then

ξ := 𝟙_{ℝ^{d}} − ρ_{r/2} ∗ 𝟙_{U_{3r/2}(F_{r})} = \frac{1}{2}𝟙_{ℝ^{d}} + ρ_{r/2} ∗ (\frac{1}{2}𝟙_{ℝ^{d}} − 𝟙_{U_{3r/2}(F_{r})})

has the above properties, and

||∇ξ||_{∞} ⩽ \frac{1}{2}||∇ρ_{r/2}||_{1} = \frac{1}{r}||∇ρ||_{1},  ||Δξ||_{∞} ⩽ \frac{1}{2}||Δρ_{r/2}||_{1} = \frac{2}{r^{2}} ||Δρ||_{1}.  (4.4)

By Lemma 2.4 we obtain ξφ ∈ D(H_{G_{r}}) and

f_{r} := (H_{G_{r}} − λ)(ξφ) = −2∇ξ · ∇φ − (Δξ)φ, spt f_{r} ⊆ spt ∇ξ ⊆ U_{2r}(F_{r}).

Then ξφ = (H_{G_{r}} − λ)^{−1}f_{r} = (H_{G_{r}} − λ)^{−1}𝟙_{U_{2r}(F_{r})}f_{r}. Since ξ = 1 on Ω \ U_{3r}(F_{r}), we can now estimate

\begin{aligned}||φ||_{1} & = ||𝟙_{U_{3r}(F_{r})}φ||_{1} + ||𝟙_{Ω\U_{3r}(F_{r})}ξφ||_{1} \\ & ⩽ ||𝟙_{U_{3r}(F_{r})}φ||_{1} + ||𝟙_{Ω\U_{3r}(F_{r})}(H_{G_{r}} − λ)^{−1}𝟙_{U_{2r}(F_{r})}||_{1→1}||f_{r}||_{1}. \end{aligned}  (4.5)

The remainder of the proof consists of estimating the terms in this pivotal inequality.

Lemma 4.1 implies

||𝟙_{U_{3r}(F_{r})}φ||_{1} ⩽ |U_{3r}(F_{r})|^{1/2}||φ||_{2} ⩽ (ω_{d}(5r)^{d}N_{t}(H))^{1/2}||φ||_{2}  (4.6)

and

||f_{r}||_{1} ⩽ |U_{2r}(F_{r})|^{1/2}||f_{r}||_{2} ⩽ (ω_{d}(4r)^{d}N_{t}(H))^{1/2}||f_{r}||_{2} ,  (4.7)
||f_{r}||_{2} ⩽ 2||∇ξ||_{∞}||∇φ||_{2} + ||Δξ||_{∞}||φ||_{2} ⩽ \frac{2}{r} ||∇ρ||_{1}\sqrt{λ} ||φ||_{2} + \frac{2}{r^{2}} ||Δρ||_{1}||φ||_{2} , (4.8)

where in (4.8) we used (4.4) and ||∇φ||_{2}^{2} = λ||φ||_{2}^{2}.

(ii) Next we estimate ||𝟙_{Ω\U_{3r}(F_{r})}(H_{G_{r}} − λ)^{−1}𝟙_{U_{2r}(F_{r})}||_{1→1}. Let δ, θ ∈ (0, 1) and

ε := δ \frac{t − λ}{t}, μ := (1 − ε)E_{0}(H_{G_{r}}) − λ.

Then (H_{G_{r}} − λ)^{−1} ⩽ ε^{−d/2}(μ − Δ)^{−1} by Theorem 3.4, and hence Proposition 3.5 implies

||𝟙_{Ω\U_{3r}(F_{r})}(H_{G_{r}} − λ)^{−1}𝟙_{U_{2r}(F_{r})}||_{1→1} ⩽ ε^{−d/2}(1 − θ^{2})^{−d/2} \frac{1}{μ}e^{−θr\sqrt{μ}}.  (4.9)

By Lemma 4.2 and the definition of ε we have

μ ⩾ (1 − ε)(t − E_{0,d}/r^{2}) − λ ⩾ t − εt − E_{0,d}/r^{2} − λ = (1 − δ)(t − λ) − E_{0,d}/r^{2}.

We now choose r such that r^{2} = \frac{c^{2}+E_{0,d}}{(1−δ)(t−λ)}, with c ⩾ d + 1 to be determined later. Then

r^{2} ⩽ \frac{7/4}{(1 − δ)(t − λ)} c^{2}  (4.10)

since E_{0,d} ⩽ \frac{3}{4}(d + 1)^{2} ⩽ \frac{3}{4}c^{2} by Lemma 4.2, and

μr^{2} ⩾ (1 − δ)(t − λ)r^{2} − E_{0,d} = c^{2}.

By (4.9) we thus obtain

||𝟙_{Ω\U_{3r}(F_{r})}(H_{G_{r}} − λ)^{−1}𝟙_{U_{2r}(F_{r})}||_{1→1} ⩽ ε^{−d/2}(1 − θ^{2})^{−d/2} \frac{r^{2}}{c^{2}} e^{−θc}.  (4.11)

(iii) In this step we incorporate an estimate for ||f_{r}||_{1} into (4.11). By (4.3) we have ||∇ρ||_{1} ⩽ c and ||Δρ||_{1} ⩽ 2c^{2}. Thus, using (4.8), (4.10) and λ < t we obtain

\frac{r^{2}}{2} ||f_{r}||_{2} ⩽ ||∇ρ||_{1}r\sqrt{λ} ||φ||_{2} + ||Δρ||_{1}||φ||_{2}

⩽ c^{2}\sqrt{\frac{7/4}{1 − δ}} · \sqrt{\frac{λ}{t − λ}} ||φ||_{2} + 2c^{2}||φ||_{2} ⩽ c^{2}C_{δ}\sqrt{\frac{t}{t − λ}} ||φ||_{2} ,

with C_{δ} = \sqrt{2/(1 − δ)} + 2. Recalling ε = δ \frac{t−λ}{t}, we infer by (4.11) that

\begin{matrix}||𝟙_{Ω\U_{3r}(F_{r})}(H_{G_{r}} − λ)^{−1}𝟙_{U_{2r}(F_{r})}||_{1→1}||f_{r}||_{2} ⩽ ε^{−d/2} & (1 − θ^{2})^{−d/2}e^{−θc} \frac{r^{2}}{c^{2}} ||f_{r}||_{2} \\ ⩽ δ^{−d/2}(\frac{t}{t−λ})^{(d+1)/2} & (1 − θ^{2})^{−d/2}e^{−θc} · 2C_{δ}||φ||_{2} . \end{matrix} (4.12)

Now we set K_{δ,θ} := \frac{5}{4}δ(1 − θ^{2}) and choose

c := \frac{d + 1}{2θ} ln (\frac{1}{K_{δ,θ}} · \frac{t}{t − λ}) .

Then

δ^{−d/2}(\frac{t}{t−λ})^{(d+1)/2}(1 − θ^{2})^{−d/2}e^{−θc} = (\frac{5}{4})^{d/2}K_{δ,θ}^{1/2},

so by (4.7) and (4.12) we obtain

\begin{matrix}||𝟙_{Ω\U_{3r}(F_{r})}(H_{G_{r}} − λ)^{−1}𝟙_{U_{2r}(F_{r})}||_{1→1}||f_{r}||_{1} \\ ⩽ (ω_{d}(5r)^{d}N_{t}(H))^{1/2} · K_{δ,θ}^{1/2} · 2C_{δ}||φ||_{2} . \end{matrix}  (4.13)

(iv) We set θ := \frac{1}{2} and δ := \frac{16}{45}, so that K_{δ,θ} = \frac{1}{3} and hence

c = (d + 1) ln \frac{3t}{t−λ} ⩾ d + 1  (4.14)

as required above. Moreover, one easily verifies that K_{δ,θ}^{1/2} · 2C_{δ} ⩽ \frac{9}{2}. By (4.5), (4.6) and (4.13) we conclude that

||φ||_{1}^{2} ⩽ (\frac{11}{2}(ω_{d}(5r)^{d}N_{t}(H))^{1/2}||φ||_{2})^{2} = (\frac{11}{2})^{2}ω_{d}(5r)^{d}N_{t}(H)||φ||_{2}^{2} .  (4.15)

Stirling’s formula yields

ω_{d} = \frac{π^{d/2}}{Γ(\frac{d}{2} + 1)} ⩽ \frac{π^{d/2}}{\sqrt{2πd/2} (\frac{d}{2e})^{d/2}} = (πd)^{−1/2}(2πe)^{d/2}d^{−d/2},

so by (4.10) we obtain

ω_{d}(5r)^{d} ⩽ (πd)^{−1/2}(2πe · \frac{7/4}{1−δ})^{d/2}d^{−d/2} · 5^{d}c^{d}(t − λ)^{−d/2}.

Using 2πe · \frac{7/4}{1−δ} ⩽ 7^{2}, (d + 1)^{d} ⩽ 2d^{d+1/2} and (4.14) we finally derive

ω_{d}(5r)^{d} ⩽ π^{−1/2}(7 · 5)^{d} · 2d^{d/2}(ln \frac{3t}{t−λ})^{d}(t − λ)^{−d/2}.

Together with (4.15) this proves the assertion since (\frac{11}{2})^{2} π^{−1/2} · 2 ⩽ 35.

References

[ArWa03] W. Arendt and M. Warma, Dirichlet and Neumann boundary conditions: What is in between? J. Evol. Equ. 3 (2003), no. 1, 119–135.

[BHV13] M. van den Berg, R. Hempel and J. Voigt, L_{1}-estimates for eigenfunctions of the Dirichlet Laplacian, to appear in J. Spectr. Theory.

[BeDa89] M. van den Berg and E. B. Davies, Heat flow out of regions in ℝ^{m}, Math. Z. 202 (1989), no. 4, 463–482.

[CoSi08] T. Coulhon and A. Sikora, Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem, Proc. Lond. Math. Soc. 96 (2008), no. 2, 507–544.

[Dav95] E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995), no. 1, 141–169.

[KLVW13] M. Keller, D. Lenz, H. Vogt and R. Wojciechowski, Note on basic features of large time behaviour of heat kernels, to appear in J.

Reine angew. Math., DOI: 10.1515/crelle-2013-0070 .

[LiYa83] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), no. 3, 309–318.

[MVV05] A. Manavi, H. Vogt and J. Voigt, Domination of semigroups associated with sectorial forms, J. Operator Theory 54 (2005), no. 1, 9–25.

[Ouh06] E. M. Ouhabaz, Comportement des noyaux de la chaleur des opérateurs de Schrödinger et applications à certaines équations paraboliques semi-linéaires, J. Funct. Anal. 238 (2006), no. 1, 278– 297.

[OuWa07] E. M. Ouhabaz and F.-Y. Wang, Sharp estimates for intrinsic ultracontractivity on C^{1,α}-domains. Manuscripta Math. 122 (2007), no. 2, 229–244.